Understanding the‘ScientificRevolution’ 145The remainder then as a general rule
Of their cube roots subtracted
Will be equal to your principal thing.
[Thenx=^3
√
u−^3√
v.]The point about the solution of the cubic (which is never now taught in schools, and hardly in
universities) is that it extended the simple reckoners’ algebra beyond its capabilities, if not for any
obviously useful purpose. One of the problems set by Fiore to Tartaglia in their 1535 contest sounds
very much in the reckoners’ tradition. However, it belongs in the category of problems which are
practical only in appearance; one cannot imagine it being the answer to a merchant’s needs.
A man sells a sapphire for five hundred ducats, making a profit of the cube root of his capital.
How much is this profit?
This is the equation ‘cube and thing equal 500’, or as we would say,x^3 +x=500.
How were such solutions written in the 1530s? Tartaglia’s exposition in his published letter
of 23 April 1539 to Cardano gives the answer, in a question which he seems to have chosen
particularly to display his ability to deal with difficulties:And if it were 1 cube plus 1 thing equal to 11, it would be necessary to find two numbers or quantities such that one
is 11 more than the other, and that the product of the one by the other should be 271 , that is the cube of the third of
the things, whence operating as above it will be found that our thing is R/ u. cube R/30 10831 plus 5^12 minus R/ u. cube
R/30 10831 minus 5^12 and not other...(Tartaglia 1959, p. 122)The ‘u.’ in the above is for ‘universal’; the whole means simply ‘cube root’. ‘R/’ is a common sign for
‘root’ at this time. We can recognize Tartaglia’s solution, in our notation, as
3√√
30
31
108
+ 5
1
2
−
3√√
30
31
108
− 5
1
2
And we can complacently note how much Tartaglia missed the use of brackets in particular, as well
as many other improvements in notation which were introduced in the next century. In any case
it seems that the arrival of formulae of this complexity meant that both the writing of algebra and
the way in which numbers themselves were thought about needed radical change; and that is what
happened. This, at any rate, is Jacob Klein’s thesis:While, however, the ‘algebra’ which has Arabic sources is continually elaborated in respect to techniques of calculation,
for instance by the introduction of ‘negative’, ‘irrational’, and even so-called ‘imaginary’ magnitudes (numeri ‘absurdi’
or ‘ficti’, ‘irrationales’ or ‘surdi’, ‘impossibiles’ or ‘sophistici’),^9 by the solution of cubic equations, and in its whole mode
of operating with numbers and number signs, its self-understanding fails to keep pace with these technical advances.
This algebraic school becomes conscious of its own ‘scientific’ character and of the novelty of its ‘number’ concept
only at the moment of direct contact with the corresponding Greek science, that is, theArithmeticaof Diophantus.
(Klein 1968, pp. 147–8)
It is probable that Klein did not know of the ‘abstract algebra’ of al-Karaj ̄i, al-Samaw‘al, and
Shar ̄af al-D ̄in al-T ̄us ̄i; and that he did know that Diophantus was available to the Islamic world. He
also seems to have given a lesser weight to the very influential introduction of decimal fractions,
which made it possible at least tothinkof approximating roots, and even numbers likeπ,as
closely as one liked. We have looked at the question of their ‘invention’ in Chapter 5; in Europe,- Each of these three pairs of Latin terms is the old equivalent of one of the modern English terms, at least approximately.